[Paper Review] A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach

Aryan Mokhtari, Asuman Ozdaglar+1 more|arXiv (Cornell University)|Jan 24, 2019

Sparse and Compressive Sensing Techniques36 references74 citations

TL;DR

The paper analyzes Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods for saddle point problems by viewing them as proximal point method approximations, deriving unified convergence results for bilinear and strongly convex-strongly concave settings, and generalizing OGDA with broader parameter choices.

ABSTRACT

In this paper we consider solving saddle point problems using two variants of Gradient Descent-Ascent algorithms, Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods. We show that both of these algorithms admit a unified analysis as approximations of the classical proximal point method for solving saddle point problems. This viewpoint enables us to develop a new framework for analyzing EG and OGDA for bilinear and strongly convex-strongly concave settings. Moreover, we use the proximal point approximation interpretation to generalize the results for OGDA for a wide range of parameters.

Motivation & Objective

• Motivate and study saddle point problems in convex-concave form and their relevance to zero-sum games, robust optimization, control, and GANs.
• Develop a unified proximal point-based framework to analyze EG and OGDA methods.
• Establish convergence rates for EG and OGDA in bilinear and strongly convex-strongly concave scenarios.
• Generalize OGDA with broader parameter choices and prove convergence for the generalized method.

Proposed method

• Model saddle point problems under bilinear and general smooth convex-concave assumptions.
• Interpret OGDA updates as approximations of the proximal point method with an o(η^2) error (Proposition 1).
• Prove linear convergence of OGDA in bilinear (Theorem 3) and strongly convex-strongly concave (Theorem 4) settings with appropriate step sizes.
• Generalize OGDA to allow unequal gradient and momentum coefficients and prove convergence under specified conditions (Theorem 5).
• Show EG updates as proximal point approximations with an error bound and establish linear convergence in bilinear (Theorem 6) and strongly convex-strongly concave cases (Theorem 7).
• Relate results to existing literature and compare rate guarantees (Table 1 in the paper).

Experimental results

Research questions

• RQ1 Can EG and OGDA be interpreted as proximal point method approximations for saddle point problems?
• RQ2 What linear convergence rates can be established for EG and OGDA in bilinear and strongly convex-strongly concave settings?
• RQ3 How does a generalized OGDA with flexible parameter choices affect convergence?
• RQ4 How does the proximal point viewpoint unify the analysis of EG and OGDA with existing results for saddle point problems?

Key findings

• OGDA converges linearly in bilinear saddle point problems when using a suitably chosen step size, with an overall iteration complexity of O(kappa log(1/epsilon)).
• OGDA converges linearly in strongly convex-strongly concave settings with a step size dependent on smoothness constants, achieving O(kappa log(1/epsilon)) iterations.
• EG converges linearly in bilinear problems and in strongly convex-strongly concave settings under explicit step-size choices, matching known optimal rates.
• A generalized OGDA with unequal coefficients for current gradient and past gradient terms remains linearly convergent under a specified range of parameters (Theorem 5).
• EG and OGDA can be regarded as proximal point method approximations with o(η^2) error, providing a unifying framework for their convergence analysis.